Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation

The paper’s Theorem 3.1 inequalities and proof steps are essentially correct and align with the candidate solution, but both omit key analytical hypotheses needed to justify the existence/regularity of the auxiliary solution φθ to the Dirichlet problem L*φθ = pθ − p on a bounded hypercube: neither explicitly assumes uniform ellipticity (or an equivalent condition) that would yield an H2 (or classical) solution and make P′θ,r well-defined; the paper even states a questionable “C2,1_per(U)” regularity for a Dirichlet problem. Apart from these gaps, the two arguments are substantially the same: both use the Green identity to derive ∫U|pθ − p|2 = ∫UL*φθ pθ and then bound Eφ |Ex∼pθ L*φ| by (i) ||Lpθ||2 in an L2-ball and (ii) a coefficient-dependent constant C in an H2-ball, exactly as in Theorem 3.1 (eqs. (17)–(19)) and its proof lines (20)–(22) , using the adjoint identity and the definition of L* .